In [1]}, we established the local conservation of mass and momentum in the low rank tensor approach with the conservative truncation (REF ). Since the associated full rank algorithm (without truncation) does not have energy conservation property, the low rank tensor scheme cannot preserve energy conservation. In fact, an implicit symplectic type time discretization is often needed for the kinetic scheme for energy conservation, e.g. see [2]}. In Figure REF , we present the time evolution of relative deviation of the total mass, total momentum and total energy of the method in [1]} for simulating the bump-on-tail instability test with truncation threshold \(\varepsilon =10^{-4}\)  (see Example REF  in Section ). It is observed that the total mass is well conserved up to the machine precision. Meanwhile, with a coarse mesh size \(32\times 64\) , it is found that the conservation error of the total momentum starts to increase at \(t=15\) , which is attributed to the boundary error as discussed in [1]}. The total energy conservation is not observed as expected.
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